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Matrix help?

  1. Oct 13, 2008 #1
    Is is possible to represent a "square matrix" with the product of the constant and identity matrix of same order of given matrix


    EX:-A=[ ] 4x4 matrix

    can I make it something like this A=(K) I 4x4

    K= constant
    I=identity matrix
     
  2. jcsd
  3. Oct 13, 2008 #2

    HallsofIvy

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    It's not clear what you are asking. If I is the identity matrix, then (K)I= K. But what doe you mean by the "constant" matrix? A= (K) I= K only if A is already the "constant" matrix itself.
     
  4. Oct 13, 2008 #3
    No. There are tons of trivial counter examples. Any square matrix with components that are *not* along the diagonal.
     
  5. Oct 13, 2008 #4
    for example
    if the matrix A is some thing like this:-
    2 0 0
    0 2 0
    0 0 2 =====> A=(2)I
    where K=2
    I=identity matrix of order 3



    Now what I want to know is if matrix A is
    a b c
    d e f
    g h i =======>A=(k)I

    is it possible to shrink A in this form for a square matrix
    how can I find what exactly k is??
     
  6. Oct 13, 2008 #5

    statdad

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    You can't do what you want if the matrix [tex] A [/tex] does not have a form like these:

    [tex]
    \begin{bmatrix} 4 & 0\\0 & 4 \end{bmatrix}, \quad \begin{bmatrix} -3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3 \end{bmatrix}
    [/tex]

    So, even more directly, if you start with

    [tex]
    \begin{bmatrix} 4 & 2\\-8 & \pi \end{bmatrix}
    [/tex]

    you cannot write this as [tex] k I_2 [/tex] no matter how imaginative you are in selecting the number [tex] k [/tex].
     
  7. Oct 13, 2008 #6

    statdad

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    I must add - this is the point both HallsOfIvy and Tac-Tics were making.
     
  8. Oct 14, 2008 #7
    yeah I got the point
    @ all people thanks for your help
     
  9. Oct 19, 2008 #8
    so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?
     
  10. Oct 20, 2008 #9

    HallsofIvy

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    Have you actually tried this multiplication? The number k times I is exactly a matrix with "k" along the diagaonal and zeros everywhere else. Why are you even asking such a question? It's a lot like asking repeatedly if 1+ 1= 3. DO it and see for yourself!
     
  11. Oct 20, 2008 #10

    statdad

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    This is an indubitable fact of mathematics.
     
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